Faraday's Law of Induction — Question 7

Question 7

A changing long-wire current links a nearby loop logarithmically

A fixed rectangular loop of height ell extends radially from a to b on one side of a long straight wire. The source current is $I(t)=I_0\cos\omega t$. Choose the loop normal along the field produced by positive I, and use $B(r)=\mu_0I/(2\pi r)$.

1. Find flux and emf.

2. Find the effect on emf amplitude of doubling both a and b while keeping ell fixed.

Question 7 - Solution

A changing long-wire current links a nearby loop logarithmically

(a) Method and calculation

$$\Phi=\ell\int_a^b\frac{\mu_0I(t)}{2\pi r}dr=\frac{\mu_0\ell I_0}{2\pi}\ln(b/a)\cos\omega t.$$$$\boxed{\mathcal E=\frac{\mu_0\ell I_0\omega}{2\pi}\ln(b/a)\sin\omega t}.$$

(b) Interpretation and further calculation

$$\ln\frac{2b}{2a}=\ln\frac ba.$$$$\boxed{\mathcal E_{0,\mathrm{new}}=\mathcal E_0}.$$

Physical check

The outward shift weakens the field but doubles the radial width; these changes compensate for this scaled geometry.